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<Title>Johnson All Pairs Shortest Paths</Title>
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<H1><A NAME="sec:johnson"></A>
<TT>johnson_all_pairs_shortest_paths</TT>
</H1>

<PRE>
<i>// named parameter version</i>
  template &lt;class VertexAndEdgeListGraph, class DistanceMatrix,
            class VertexID, class Weight, class BinaryPredicate,
            class BinaryFunction, class Infinity, class DistanceZero&gt;
  bool
  johnson_all_pairs_shortest_paths(VertexAndEdgeListGraph& g1,
               DistanceMatrix&amp; D,
               VertexID id1, Weight w1, const BinaryPredicate&amp; compare,
               const BinaryFunction&amp; combine, const Infinity&amp; inf,
               DistanceZero zero);

template &lt;typename Graph, typename DistanceMatrix, typename P, typename T, typename R&gt;
bool johnson_all_pairs_shortest_paths(Graph&amp; g, DistanceMatrix&amp; D,
  const bgl_named_params&lt;P, T, R&gt;&amp; params = <i>all defaults</i>)

<i>// non-named parameter version</i>
template &lt;typename Graph, typename DistanceMatrix,
          typename VertexIndex, typename WeightMap, typename DT&gt;
bool
johnson_all_pairs_shortest_paths(VertexAndEdgeListGraph&amp; g1,
  DistanceMatrix&amp; D,
  VertexIndex i_map, WeightMap w_map, DT zero)
</PRE>

<P>
This algorithm finds the shortest distance between every pair of
vertices in the graph. The algorithm returns false if there is a
negative weight cycle in the graph and true otherwise. The distance
between each pair of vertices is stored in the distance matrix
<TT>D</TT>. This is one of the more time intensive graph algorithms,
having a time complexity of <i>O(V E log V)</i>.

<P>This algorithm should be used to compute shortest paths between
every pair of vertices for sparse graphs. For dense graphs, use <a
href="floyd_warshall_shortest.html"><code>floyd_warshall_all_pairs_shortest_paths</code></a>.

<H3>Where Defined</H3>

<P>
<a href="../../../boost/graph/johnson_all_pairs_shortest.hpp"><TT>boost/graph/johnson_all_pairs_shortest.hpp</TT></a>


<h3>Parameters</h3>

IN: <tt>Graph&amp; g</tt>
<blockquote>
A directed or undirected graph. The graph type must be a model of
<a href="VertexListGraph.html">Vertex List Graph</a>,
<a href="EdgeListGraph.html">Edge List Graph</a>,
and <a href="IncidenceGraph.html">Incidence Graph</a>.
</blockquote>

OUT: <tt>DistanceMatrix&amp; D</tt>
<blockquote>
The length of the shortest path between each pair of vertices
<i>u,v</i> in the graph is stored in <tt>D[u][v]</tt>. The tuple of
types (<tt>DistanceMatrix, vertices_size_type, D</tt>) must be a model
of <a href="BasicMatrix.html">BasicMatrix</a> where <tt>D</tt> is the
value type of the <tt>DistanceMap</tt>.  There must be implicit conversions
between the value type of the distance matrix and the value type of the weight
map.
</blockquote>


<h3>Named Parameters</h3>

IN: <tt>weight_map(WeightMap w_map)</tt>
<blockquote>
  The weight or "length" of each edge in the graph.
  The type <tt>WeightMap</tt> must be a model of
  <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The edge descriptor type of
  the graph needs to be usable as the key type for the weight
  map.  The value type of the weight map must support a subtraction operation.<br>
  <b>Default:</b>  <tt>get(edge_weight, g)</tt>
</blockquote>

UTIL: <tt>weight_map2(WeightMap2 w2_map)</tt>
<blockquote>
  This parameter is no longer needed, and will be ignored.
</blockquote>

IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt>
<blockquote>
  This maps each vertex to an integer in the range <tt>[0,
    num_vertices(g))</tt>. This is necessary for efficient updates of the
  heap data structure in the internal call to Dijkstra's algorithm.  The type
  <tt>VertexIndexMap</tt> must be a model of
  <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The value type of the map must be an
  integer type. The vertex descriptor type of the graph needs to be
  usable as the key type of the map.<br>
  <b>Default:</b> <tt>get(vertex_index, g)</tt>
    Note: if you use this default, make sure your graph has
    an internal <tt>vertex_index</tt> property. For example,
    <tt>adjacency_list</tt> with <tt>VertexList=listS</tt> does
    not have an internal <tt>vertex_index</tt> property.
   <br>
</blockquote>


UTIL: <tt>distance_map(DistanceMap d_map)</tt>
<blockquote>
  This parameter is no longer needed, and will be ignored.
</blockquote>

IN: <tt>distance_compare(CompareFunction cmp)</tt>
<blockquote>
  This function is use to compare distances to determine
  which vertex is closer to the source vertex.
  The <tt>CompareFunction</tt> type must be a model of
  <a href="http://www.boost.org/sgi/stl/BinaryPredicate.html">Binary Predicate</a>
  and have argument types that
  match the value type of the <tt>WeightMap</tt> property map.<br>
  <b>Default:</b> <tt>std::less&lt;DT&gt;</tt> with
  <tt>DT=typename&nbsp;property_traits&lt;WeightMap&gt;::value_type</tt>
</blockquote>

IN: <tt>distance_combine(CombineFunction cmb)</tt>
<blockquote>
  This function is used to combine distances to compute the distance
  of a path. The <tt>CombineFunction</tt> type must be a model of <a
  href="http://www.boost.org/sgi/stl/BinaryFunction.html">Binary
  Function</a>. Both argument types and the return type of the binary function
  must match the value type of the <tt>WeightMap</tt> property map.  This operation is required to act as the sum operation for the weight type; in particular, it must be the inverse of the binary <tt>-</tt> operator on that type.<br>
  <b>Default:</b> <tt>std::plus&lt;DT&gt;</tt> with
   <tt>DT=typename&nbsp;property_traits&lt;WeightMap&gt;::value_type</tt>
</blockquote>

IN: <tt>distance_inf(DT inf)</tt>
<blockquote>
  This value is used to initialize the distance for each
  vertex before the start of the algorithm.
  The type <tt>DT</tt> must be the value type of the <tt>WeightMap</tt>.<br>
  <b>Default:</b> <tt>std::numeric_limits::max()</tt>
</blockquote>

IN: <tt>distance_zero(DT zero)</tt>
<blockquote>
  This value is used to initialize the distance for the source
  vertex before the start of the algorithm.  The type <tt>DT</tt>
  must be the value type of the <tt>WeightMap</tt>.<br>
  <b>Default:</b> <tt>0</tt>
</blockquote>

UTIL/OUT: <tt>color_map(ColorMap c_map)</tt>
<blockquote>
  This is used during the execution of the algorithm to mark the
  vertices. The vertices start out white and become gray when they are
  inserted in the queue. They then turn black when they are removed
  from the queue. At the end of the algorithm, vertices reachable from
  the source vertex will have been colored black. All other vertices
  will still be white. The type <tt>ColorMap</tt> must be a model of
  <a href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
  Property Map</a>. A vertex descriptor must be usable as the key type
  of the map, and the value type of the map must be a model of
  <a href="./ColorValue.html">Color Value</a>.<br>
  <b>Default:</b> an <a
  href="../../property_map/doc/iterator_property_map.html">
  <tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt>
  of <tt>default_color_type</tt> of size <tt>num_vertices(g)</tt> and
  using the <tt>i_map</tt> for the index map.
</blockquote>


<h3>Complexity</h3>

The time complexity is <i>O(V E log V)</i>.



<H3>Example</H3>

<P>
The file <a
href="../example/johnson-eg.cpp"><tt>example/johnson-eg.cpp</tt></a> applies
Johnson's algorithm for all-pairs shortest paths to the example graph
from page 568 of the CLR&nbsp;[<A
HREF="bibliography.html#clr90">8</A>].


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<TD nowrap>Copyright &copy; 2000-2001</TD><TD>
<A HREF="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</A>, Indiana University (<A HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)
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